How to identify given number is perfect square or not?

• Last digit of a perfect square is NEVER 2,3,7 or 8 And it's always 0,1,4,5,6 and 9.
e.g: 100, 81, 64, 25, 36, 529...

• If last digit of square is 5, then the 2nd last digit is always 2.
e.g: 25, 625, 1225...

• If last digit of square is 6, then the 2nd last digit is always ODD.
e.g: 36, 256...

• If last digit of a square is 1,4 and 9 then 2nd last digit is always EVEN.
e.g: 81, 121, 144, 484, 49, 1089...

• The number of zero's at the end of a square is always EVEN & the non zero part should be a perfect Square.
e.g: 900, 25600, 10000...

• 7744 is the Only Perfect square number of the form/type XXYY.
i.e 88^2 = 7744.

Cube And Cube Roots:
You should know the Cubes of numbers from 1-10.
1^3- 1
2^3- 8
3^3- 27
4^3- 64
5^3- 125
6^3- 216
7^3- 343
8^3- 512
9^3- 729
10^3- 1000

You can easily find cube of any number by using formulas given:

(a + b)^3 = a^3 + 3a^2b + 3ab^2+ b^3
(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3

Example:

23^3 = (20 + 3)³

Step 1 — Apply Formula

(20 + 3)^3 = 20^3 + 3(20^2×3) + 3(20×3^2) + 3^3

Step 2 — Solve
= 8000 + 3(400×3) + 3(20×9) + 27
= 8000 + 3600 + 540 + 27 
= 12,167

7³ = (10 − 3)³

Step 1 — Apply Formula

(10 - 3)^3 = 10^3 - 3(10^2×3) + 3(10×3^2) - 3^3

Step 2 — Solve

= 1000 - 3(100×3) + 3(10×9) - 27
= 1000 - 900 + 270 - 27 
= 343

How to identify cube of number: There is trick:
Check last digit of number and Check last digit of answer also

Number Cube Last digit
1 1 1
2 8 8
3 27 7
4 64 4
5 125 5
6 216 6
7 343 3
8 512 2
9 729 9
10 1000 0

By checking only last digits you can eliminate at least 1 or 2 or 3 options.
You will be close to answer.

Here it is expected that you should check the last digit...
As 4 is there at last in 54... According to shortcut...
There should be 4×4×4 = 16×4 i.e. 6×4= 24 i.e. 4 should be there at last digit in answer.

As there are two such options.
We will go for formula....

54³= (50+4)³
=12500+3×2500×4+3×50×16+64
=125000+30000+2400+64
=1,57,464

LCM (Least Common Multiple)

Definition →The smallest number that is divisible by all given numbers.

Steps to Find LCM
Method 1 – Prime Factorization

• Factorize each number into prime numbers.

• Take each prime factor only once with its highest power.

• Multiply them together.

Example:
Find LCM of 12 and 18.

12 = 2² × 3¹
18 = 2¹ × 3²

LCM = 2² × 3² = 36

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Method 2 – Division Method (Shortcut)

• Divide the numbers by prime numbers till all become 1.

• Multiply all divisors together.

Example:
Find LCM of 15, 20, 30.

Divisor: 15 , 20, 30

2- 15 10 15
2- 15 5 15
3- 5 5 5
5- 1 1 1

LCM = 2 × 2 × 3 × 5 = 60

HCF (Highest Common Factor)
It is also called GCD (Greatest Common Divisor).

#Definition
The HCF of two or more numbers is the largest positive integer that divides each of the numbers completely (without leaving any remainder).

Steps to Find HCF:

Method 1: Prime Factorization

1. Find the prime factors of each number.

2. Identify the common prime factors.

3. Multiply them to get the HCF.

Example:
Find HCF of 18 and 24.

Prime factors of 18 → 2×3×3
Prime factors of 24 → 2×2×2×3

Common factors → 2×3-> HCF = 6

Method 2: Division Method (Euclidean Algorithm)

1. Divide the larger number by the smaller number.

2. Take the remainder.

3. Divide the smaller number by the remainder.

4. Repeat until the remainder becomes 0.
The last divisor is the HCF.

Example:
Find HCF of 36 and 60:

60/36=1 remainder 24
36/24=1 remainder 12
24/12=2 remainder 0

HCF = 12

Another way to find HCF when Two numbers are there and LCM is Given :

Formula: HCF × LCM = N1 × N2

e.g: N1 = 15, N2 = 20, LCM = 60, HCF= ?

HCF = 15×20/60
= 300/60
HCF = 5


Note: This formula is valid for only two numbers.

Face value And Place value
1. Face Value
The face value of a digit is the actual value of the digit itself, irrespective of its position in the number.

#Example
For the number 4725:
Face value of 4 = 4
Face value of 7 = 7
Face value of 2 = 2
Face value of 5 = 5
✅ Face value = digit itself

2. Place Value
It is the place at which Digit located like 10th position, 100th position etc.

#Example:
For the number 4725:
Place value of 4 = 4000
Place value of 7 = 700
Place value of 2 = 20
Place value of 5 = 5

Quick Trick to Remember
Face value → What you see 👀
Place value → Where it is 📍